We can calculate the angles of the triangle using the Law of Sines and Cosines.
The following parameters are provided for the triangle:
[tex]\begin{gathered} a=45 \\ b=32 \\ c=24 \end{gathered}[/tex]Measure of ∠A
The law of cosines can be applied as follows:
[tex]\begin{gathered} a^2=b^2+c^2-2bc\cos A \\ \therefore \\ A=\arccos(\frac{b^2+c^2-a^2}{2bc}) \end{gathered}[/tex]Substituting known values, we have:
[tex]\begin{gathered} A=\arccos(\frac{32^2+24^2-45^2}{2\times32\times24}) \\ A=106.1\degree \end{gathered}[/tex]Measure of ∠B
We can apply the law of sines as follows:
[tex]\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B} \\ \therefore \\ B=\arcsin(\frac{b\sin A}{a}) \end{gathered}[/tex]Substituting known values, we have:
[tex]\begin{gathered} B=\arcsin(\frac{32\times\sin106.1}{45}) \\ B=43.1\degree \end{gathered}[/tex]Measure of ∠C
The sum of angles in a triangle is 180 degrees. Therefore, the measure of angle C is:
[tex]\begin{gathered} C=180-A-B \\ C=180-106.1-43.1 \\ C=30.8\degree \end{gathered}[/tex]ANSWERS
[tex]\begin{gathered} m\angle A=106.1\operatorname{\degree} \\ m\angle B=43.1\operatorname{\degree} \\ m\angle C=30.8\operatorname{\degree} \end{gathered}[/tex]